Properties

Label 8036.2.a.j.1.4
Level $8036$
Weight $2$
Character 8036.1
Self dual yes
Analytic conductor $64.168$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8036,2,Mod(1,8036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.1935333.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 13x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1148)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.43444\) of defining polynomial
Character \(\chi\) \(=\) 8036.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.43444 q^{3} -1.63444 q^{5} -0.942381 q^{9} +O(q^{10})\) \(q+1.43444 q^{3} -1.63444 q^{5} -0.942381 q^{9} +3.97894 q^{11} +2.47100 q^{13} -2.34450 q^{15} -4.38106 q^{17} +0.0365556 q^{19} +0.565560 q^{23} -2.32862 q^{25} -5.65511 q^{27} +2.34450 q^{29} -0.365564 q^{31} +5.70755 q^{33} -1.76093 q^{37} +3.54450 q^{39} -1.00000 q^{41} -2.38106 q^{43} +1.54026 q^{45} -13.3188 q^{47} -6.28436 q^{51} -1.63444 q^{53} -6.50332 q^{55} +0.0524368 q^{57} +5.70755 q^{59} -12.6362 q^{61} -4.03868 q^{65} +14.1764 q^{67} +0.811262 q^{69} +7.10081 q^{71} +5.24781 q^{73} -3.34027 q^{75} +2.14700 q^{79} -5.28478 q^{81} -13.1008 q^{83} +7.16056 q^{85} +3.36305 q^{87} +1.33073 q^{89} -0.524380 q^{93} -0.0597478 q^{95} -6.30332 q^{97} -3.74967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{3} - 3 q^{5} + 5 q^{9} - 7 q^{13} + 3 q^{15} + 3 q^{17} - 10 q^{19} + 12 q^{23} + 2 q^{25} - 14 q^{27} - 3 q^{29} - 7 q^{31} + 3 q^{33} + q^{37} + 7 q^{39} - 5 q^{41} + 13 q^{43} + 3 q^{45} - 9 q^{47} + 9 q^{51} - 3 q^{53} - 9 q^{55} - 11 q^{57} + 3 q^{59} - 16 q^{61} + 3 q^{65} + 19 q^{67} - 24 q^{69} - 12 q^{71} - 4 q^{73} - 8 q^{75} + 28 q^{79} + 5 q^{81} - 18 q^{83} - 15 q^{85} + 6 q^{87} + q^{93} + 3 q^{95} - 4 q^{97} - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.43444 0.828175 0.414087 0.910237i \(-0.364101\pi\)
0.414087 + 0.910237i \(0.364101\pi\)
\(4\) 0 0
\(5\) −1.63444 −0.730942 −0.365471 0.930823i \(-0.619092\pi\)
−0.365471 + 0.930823i \(0.619092\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.942381 −0.314127
\(10\) 0 0
\(11\) 3.97894 1.19969 0.599847 0.800115i \(-0.295228\pi\)
0.599847 + 0.800115i \(0.295228\pi\)
\(12\) 0 0
\(13\) 2.47100 0.685331 0.342665 0.939457i \(-0.388670\pi\)
0.342665 + 0.939457i \(0.388670\pi\)
\(14\) 0 0
\(15\) −2.34450 −0.605347
\(16\) 0 0
\(17\) −4.38106 −1.06256 −0.531281 0.847196i \(-0.678289\pi\)
−0.531281 + 0.847196i \(0.678289\pi\)
\(18\) 0 0
\(19\) 0.0365556 0.00838643 0.00419321 0.999991i \(-0.498665\pi\)
0.00419321 + 0.999991i \(0.498665\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.565560 0.117927 0.0589637 0.998260i \(-0.481220\pi\)
0.0589637 + 0.998260i \(0.481220\pi\)
\(24\) 0 0
\(25\) −2.32862 −0.465724
\(26\) 0 0
\(27\) −5.65511 −1.08833
\(28\) 0 0
\(29\) 2.34450 0.435363 0.217681 0.976020i \(-0.430151\pi\)
0.217681 + 0.976020i \(0.430151\pi\)
\(30\) 0 0
\(31\) −0.365564 −0.0656573 −0.0328286 0.999461i \(-0.510452\pi\)
−0.0328286 + 0.999461i \(0.510452\pi\)
\(32\) 0 0
\(33\) 5.70755 0.993556
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.76093 −0.289495 −0.144748 0.989469i \(-0.546237\pi\)
−0.144748 + 0.989469i \(0.546237\pi\)
\(38\) 0 0
\(39\) 3.54450 0.567574
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −2.38106 −0.363108 −0.181554 0.983381i \(-0.558113\pi\)
−0.181554 + 0.983381i \(0.558113\pi\)
\(44\) 0 0
\(45\) 1.54026 0.229609
\(46\) 0 0
\(47\) −13.3188 −1.94275 −0.971374 0.237554i \(-0.923654\pi\)
−0.971374 + 0.237554i \(0.923654\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.28436 −0.879987
\(52\) 0 0
\(53\) −1.63444 −0.224507 −0.112254 0.993680i \(-0.535807\pi\)
−0.112254 + 0.993680i \(0.535807\pi\)
\(54\) 0 0
\(55\) −6.50332 −0.876907
\(56\) 0 0
\(57\) 0.0524368 0.00694543
\(58\) 0 0
\(59\) 5.70755 0.743059 0.371530 0.928421i \(-0.378833\pi\)
0.371530 + 0.928421i \(0.378833\pi\)
\(60\) 0 0
\(61\) −12.6362 −1.61790 −0.808948 0.587880i \(-0.799963\pi\)
−0.808948 + 0.587880i \(0.799963\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.03868 −0.500937
\(66\) 0 0
\(67\) 14.1764 1.73193 0.865964 0.500107i \(-0.166706\pi\)
0.865964 + 0.500107i \(0.166706\pi\)
\(68\) 0 0
\(69\) 0.811262 0.0976644
\(70\) 0 0
\(71\) 7.10081 0.842711 0.421355 0.906896i \(-0.361554\pi\)
0.421355 + 0.906896i \(0.361554\pi\)
\(72\) 0 0
\(73\) 5.24781 0.614209 0.307105 0.951676i \(-0.400640\pi\)
0.307105 + 0.951676i \(0.400640\pi\)
\(74\) 0 0
\(75\) −3.34027 −0.385701
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.14700 0.241556 0.120778 0.992680i \(-0.461461\pi\)
0.120778 + 0.992680i \(0.461461\pi\)
\(80\) 0 0
\(81\) −5.28478 −0.587197
\(82\) 0 0
\(83\) −13.1008 −1.43800 −0.719000 0.695010i \(-0.755400\pi\)
−0.719000 + 0.695010i \(0.755400\pi\)
\(84\) 0 0
\(85\) 7.16056 0.776671
\(86\) 0 0
\(87\) 3.36305 0.360556
\(88\) 0 0
\(89\) 1.33073 0.141057 0.0705283 0.997510i \(-0.477531\pi\)
0.0705283 + 0.997510i \(0.477531\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −0.524380 −0.0543757
\(94\) 0 0
\(95\) −0.0597478 −0.00612999
\(96\) 0 0
\(97\) −6.30332 −0.640005 −0.320003 0.947417i \(-0.603684\pi\)
−0.320003 + 0.947417i \(0.603684\pi\)
\(98\) 0 0
\(99\) −3.74967 −0.376856
\(100\) 0 0
\(101\) 10.0637 1.00137 0.500687 0.865628i \(-0.333081\pi\)
0.500687 + 0.865628i \(0.333081\pi\)
\(102\) 0 0
\(103\) −17.1713 −1.69193 −0.845967 0.533235i \(-0.820976\pi\)
−0.845967 + 0.533235i \(0.820976\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.45088 0.526956 0.263478 0.964665i \(-0.415130\pi\)
0.263478 + 0.964665i \(0.415130\pi\)
\(108\) 0 0
\(109\) −7.55788 −0.723914 −0.361957 0.932195i \(-0.617891\pi\)
−0.361957 + 0.932195i \(0.617891\pi\)
\(110\) 0 0
\(111\) −2.52595 −0.239753
\(112\) 0 0
\(113\) −1.75643 −0.165231 −0.0826154 0.996582i \(-0.526327\pi\)
−0.0826154 + 0.996582i \(0.526327\pi\)
\(114\) 0 0
\(115\) −0.924371 −0.0861980
\(116\) 0 0
\(117\) −2.32862 −0.215281
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.83194 0.439267
\(122\) 0 0
\(123\) −1.43444 −0.129339
\(124\) 0 0
\(125\) 11.9782 1.07136
\(126\) 0 0
\(127\) 3.12398 0.277208 0.138604 0.990348i \(-0.455738\pi\)
0.138604 + 0.990348i \(0.455738\pi\)
\(128\) 0 0
\(129\) −3.41548 −0.300717
\(130\) 0 0
\(131\) −9.15630 −0.799989 −0.399995 0.916517i \(-0.630988\pi\)
−0.399995 + 0.916517i \(0.630988\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.24291 0.795503
\(136\) 0 0
\(137\) −17.9966 −1.53755 −0.768775 0.639520i \(-0.779133\pi\)
−0.768775 + 0.639520i \(0.779133\pi\)
\(138\) 0 0
\(139\) −12.9527 −1.09863 −0.549316 0.835614i \(-0.685112\pi\)
−0.549316 + 0.835614i \(0.685112\pi\)
\(140\) 0 0
\(141\) −19.1050 −1.60893
\(142\) 0 0
\(143\) 9.83194 0.822188
\(144\) 0 0
\(145\) −3.83194 −0.318225
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.2999 1.25341 0.626707 0.779255i \(-0.284402\pi\)
0.626707 + 0.779255i \(0.284402\pi\)
\(150\) 0 0
\(151\) −20.4007 −1.66018 −0.830091 0.557628i \(-0.811712\pi\)
−0.830091 + 0.557628i \(0.811712\pi\)
\(152\) 0 0
\(153\) 4.12862 0.333779
\(154\) 0 0
\(155\) 0.597491 0.0479916
\(156\) 0 0
\(157\) 10.1081 0.806716 0.403358 0.915042i \(-0.367843\pi\)
0.403358 + 0.915042i \(0.367843\pi\)
\(158\) 0 0
\(159\) −2.34450 −0.185931
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.14940 −0.168354 −0.0841769 0.996451i \(-0.526826\pi\)
−0.0841769 + 0.996451i \(0.526826\pi\)
\(164\) 0 0
\(165\) −9.32862 −0.726232
\(166\) 0 0
\(167\) 1.36092 0.105311 0.0526555 0.998613i \(-0.483231\pi\)
0.0526555 + 0.998613i \(0.483231\pi\)
\(168\) 0 0
\(169\) −6.89418 −0.530321
\(170\) 0 0
\(171\) −0.0344493 −0.00263440
\(172\) 0 0
\(173\) 10.1485 0.771572 0.385786 0.922588i \(-0.373930\pi\)
0.385786 + 0.922588i \(0.373930\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.18714 0.615383
\(178\) 0 0
\(179\) −7.18806 −0.537261 −0.268630 0.963243i \(-0.586571\pi\)
−0.268630 + 0.963243i \(0.586571\pi\)
\(180\) 0 0
\(181\) −7.94623 −0.590638 −0.295319 0.955399i \(-0.595426\pi\)
−0.295319 + 0.955399i \(0.595426\pi\)
\(182\) 0 0
\(183\) −18.1258 −1.33990
\(184\) 0 0
\(185\) 2.87813 0.211604
\(186\) 0 0
\(187\) −17.4319 −1.27475
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.2376 −1.60906 −0.804529 0.593913i \(-0.797582\pi\)
−0.804529 + 0.593913i \(0.797582\pi\)
\(192\) 0 0
\(193\) 8.92940 0.642752 0.321376 0.946952i \(-0.395855\pi\)
0.321376 + 0.946952i \(0.395855\pi\)
\(194\) 0 0
\(195\) −5.79325 −0.414863
\(196\) 0 0
\(197\) 9.70043 0.691127 0.345563 0.938395i \(-0.387688\pi\)
0.345563 + 0.938395i \(0.387688\pi\)
\(198\) 0 0
\(199\) 9.80451 0.695023 0.347512 0.937676i \(-0.387027\pi\)
0.347512 + 0.937676i \(0.387027\pi\)
\(200\) 0 0
\(201\) 20.3353 1.43434
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.63444 0.114154
\(206\) 0 0
\(207\) −0.532973 −0.0370442
\(208\) 0 0
\(209\) 0.145452 0.0100612
\(210\) 0 0
\(211\) −16.3395 −1.12486 −0.562428 0.826847i \(-0.690132\pi\)
−0.562428 + 0.826847i \(0.690132\pi\)
\(212\) 0 0
\(213\) 10.1857 0.697912
\(214\) 0 0
\(215\) 3.89168 0.265411
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 7.52767 0.508673
\(220\) 0 0
\(221\) −10.8256 −0.728207
\(222\) 0 0
\(223\) 16.9703 1.13642 0.568208 0.822885i \(-0.307637\pi\)
0.568208 + 0.822885i \(0.307637\pi\)
\(224\) 0 0
\(225\) 2.19445 0.146296
\(226\) 0 0
\(227\) 23.9782 1.59149 0.795743 0.605634i \(-0.207081\pi\)
0.795743 + 0.605634i \(0.207081\pi\)
\(228\) 0 0
\(229\) 5.07735 0.335520 0.167760 0.985828i \(-0.446347\pi\)
0.167760 + 0.985828i \(0.446347\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.2293 −0.801166 −0.400583 0.916261i \(-0.631192\pi\)
−0.400583 + 0.916261i \(0.631192\pi\)
\(234\) 0 0
\(235\) 21.7687 1.42004
\(236\) 0 0
\(237\) 3.07974 0.200051
\(238\) 0 0
\(239\) −17.0981 −1.10599 −0.552993 0.833186i \(-0.686515\pi\)
−0.552993 + 0.833186i \(0.686515\pi\)
\(240\) 0 0
\(241\) −6.61085 −0.425843 −0.212921 0.977069i \(-0.568298\pi\)
−0.212921 + 0.977069i \(0.568298\pi\)
\(242\) 0 0
\(243\) 9.38464 0.602025
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.0903287 0.00574748
\(248\) 0 0
\(249\) −18.7923 −1.19092
\(250\) 0 0
\(251\) −14.5209 −0.916553 −0.458277 0.888810i \(-0.651533\pi\)
−0.458277 + 0.888810i \(0.651533\pi\)
\(252\) 0 0
\(253\) 2.25033 0.141477
\(254\) 0 0
\(255\) 10.2714 0.643219
\(256\) 0 0
\(257\) 3.78315 0.235987 0.117993 0.993014i \(-0.462354\pi\)
0.117993 + 0.993014i \(0.462354\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.20941 −0.136759
\(262\) 0 0
\(263\) −30.5159 −1.88169 −0.940846 0.338836i \(-0.889967\pi\)
−0.940846 + 0.338836i \(0.889967\pi\)
\(264\) 0 0
\(265\) 2.67138 0.164102
\(266\) 0 0
\(267\) 1.90885 0.116820
\(268\) 0 0
\(269\) 4.04624 0.246704 0.123352 0.992363i \(-0.460636\pi\)
0.123352 + 0.992363i \(0.460636\pi\)
\(270\) 0 0
\(271\) 0.130195 0.00790876 0.00395438 0.999992i \(-0.498741\pi\)
0.00395438 + 0.999992i \(0.498741\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.26543 −0.558726
\(276\) 0 0
\(277\) 27.3964 1.64609 0.823046 0.567975i \(-0.192273\pi\)
0.823046 + 0.567975i \(0.192273\pi\)
\(278\) 0 0
\(279\) 0.344501 0.0206247
\(280\) 0 0
\(281\) −17.9472 −1.07064 −0.535319 0.844650i \(-0.679809\pi\)
−0.535319 + 0.844650i \(0.679809\pi\)
\(282\) 0 0
\(283\) −19.8925 −1.18248 −0.591242 0.806494i \(-0.701362\pi\)
−0.591242 + 0.806494i \(0.701362\pi\)
\(284\) 0 0
\(285\) −0.0857046 −0.00507670
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.19365 0.129038
\(290\) 0 0
\(291\) −9.04174 −0.530036
\(292\) 0 0
\(293\) −9.76308 −0.570365 −0.285183 0.958473i \(-0.592054\pi\)
−0.285183 + 0.958473i \(0.592054\pi\)
\(294\) 0 0
\(295\) −9.32862 −0.543133
\(296\) 0 0
\(297\) −22.5013 −1.30566
\(298\) 0 0
\(299\) 1.39750 0.0808193
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 14.4358 0.829313
\(304\) 0 0
\(305\) 20.6530 1.18259
\(306\) 0 0
\(307\) −0.571400 −0.0326115 −0.0163058 0.999867i \(-0.505191\pi\)
−0.0163058 + 0.999867i \(0.505191\pi\)
\(308\) 0 0
\(309\) −24.6311 −1.40122
\(310\) 0 0
\(311\) −18.6256 −1.05616 −0.528080 0.849195i \(-0.677088\pi\)
−0.528080 + 0.849195i \(0.677088\pi\)
\(312\) 0 0
\(313\) −28.6875 −1.62152 −0.810758 0.585382i \(-0.800945\pi\)
−0.810758 + 0.585382i \(0.800945\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.5125 0.815101 0.407551 0.913183i \(-0.366383\pi\)
0.407551 + 0.913183i \(0.366383\pi\)
\(318\) 0 0
\(319\) 9.32862 0.522302
\(320\) 0 0
\(321\) 7.81896 0.436412
\(322\) 0 0
\(323\) −0.160152 −0.00891110
\(324\) 0 0
\(325\) −5.75401 −0.319175
\(326\) 0 0
\(327\) −10.8413 −0.599527
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −19.4522 −1.06919 −0.534596 0.845108i \(-0.679536\pi\)
−0.534596 + 0.845108i \(0.679536\pi\)
\(332\) 0 0
\(333\) 1.65947 0.0909383
\(334\) 0 0
\(335\) −23.1705 −1.26594
\(336\) 0 0
\(337\) −20.6956 −1.12736 −0.563681 0.825993i \(-0.690615\pi\)
−0.563681 + 0.825993i \(0.690615\pi\)
\(338\) 0 0
\(339\) −2.51949 −0.136840
\(340\) 0 0
\(341\) −1.45456 −0.0787687
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.32596 −0.0713870
\(346\) 0 0
\(347\) 6.83712 0.367036 0.183518 0.983016i \(-0.441251\pi\)
0.183518 + 0.983016i \(0.441251\pi\)
\(348\) 0 0
\(349\) −5.72280 −0.306335 −0.153167 0.988200i \(-0.548947\pi\)
−0.153167 + 0.988200i \(0.548947\pi\)
\(350\) 0 0
\(351\) −13.9738 −0.745864
\(352\) 0 0
\(353\) −11.2493 −0.598738 −0.299369 0.954137i \(-0.596776\pi\)
−0.299369 + 0.954137i \(0.596776\pi\)
\(354\) 0 0
\(355\) −11.6058 −0.615973
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.601190 −0.0317296 −0.0158648 0.999874i \(-0.505050\pi\)
−0.0158648 + 0.999874i \(0.505050\pi\)
\(360\) 0 0
\(361\) −18.9987 −0.999930
\(362\) 0 0
\(363\) 6.93112 0.363790
\(364\) 0 0
\(365\) −8.57721 −0.448951
\(366\) 0 0
\(367\) −21.1077 −1.10181 −0.550907 0.834566i \(-0.685718\pi\)
−0.550907 + 0.834566i \(0.685718\pi\)
\(368\) 0 0
\(369\) 0.942381 0.0490584
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −26.8297 −1.38919 −0.694593 0.719403i \(-0.744416\pi\)
−0.694593 + 0.719403i \(0.744416\pi\)
\(374\) 0 0
\(375\) 17.1820 0.887272
\(376\) 0 0
\(377\) 5.79325 0.298368
\(378\) 0 0
\(379\) −12.6585 −0.650225 −0.325112 0.945675i \(-0.605402\pi\)
−0.325112 + 0.945675i \(0.605402\pi\)
\(380\) 0 0
\(381\) 4.48116 0.229577
\(382\) 0 0
\(383\) −6.49545 −0.331902 −0.165951 0.986134i \(-0.553069\pi\)
−0.165951 + 0.986134i \(0.553069\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.24386 0.114062
\(388\) 0 0
\(389\) −18.3163 −0.928673 −0.464337 0.885659i \(-0.653707\pi\)
−0.464337 + 0.885659i \(0.653707\pi\)
\(390\) 0 0
\(391\) −2.47775 −0.125305
\(392\) 0 0
\(393\) −13.1342 −0.662531
\(394\) 0 0
\(395\) −3.50913 −0.176564
\(396\) 0 0
\(397\) −5.71680 −0.286918 −0.143459 0.989656i \(-0.545822\pi\)
−0.143459 + 0.989656i \(0.545822\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.6987 0.534268 0.267134 0.963659i \(-0.413923\pi\)
0.267134 + 0.963659i \(0.413923\pi\)
\(402\) 0 0
\(403\) −0.903308 −0.0449970
\(404\) 0 0
\(405\) 8.63763 0.429207
\(406\) 0 0
\(407\) −7.00663 −0.347306
\(408\) 0 0
\(409\) 32.9005 1.62683 0.813413 0.581686i \(-0.197607\pi\)
0.813413 + 0.581686i \(0.197607\pi\)
\(410\) 0 0
\(411\) −25.8150 −1.27336
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 21.4124 1.05109
\(416\) 0 0
\(417\) −18.5799 −0.909860
\(418\) 0 0
\(419\) 7.12676 0.348165 0.174083 0.984731i \(-0.444304\pi\)
0.174083 + 0.984731i \(0.444304\pi\)
\(420\) 0 0
\(421\) 16.3900 0.798798 0.399399 0.916777i \(-0.369219\pi\)
0.399399 + 0.916777i \(0.369219\pi\)
\(422\) 0 0
\(423\) 12.5514 0.610270
\(424\) 0 0
\(425\) 10.2018 0.494861
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 14.1033 0.680915
\(430\) 0 0
\(431\) −1.33942 −0.0645174 −0.0322587 0.999480i \(-0.510270\pi\)
−0.0322587 + 0.999480i \(0.510270\pi\)
\(432\) 0 0
\(433\) −9.72105 −0.467164 −0.233582 0.972337i \(-0.575045\pi\)
−0.233582 + 0.972337i \(0.575045\pi\)
\(434\) 0 0
\(435\) −5.49668 −0.263546
\(436\) 0 0
\(437\) 0.0206744 0.000988989 0
\(438\) 0 0
\(439\) −13.6772 −0.652775 −0.326388 0.945236i \(-0.605831\pi\)
−0.326388 + 0.945236i \(0.605831\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.4616 1.30474 0.652370 0.757901i \(-0.273775\pi\)
0.652370 + 0.757901i \(0.273775\pi\)
\(444\) 0 0
\(445\) −2.17499 −0.103104
\(446\) 0 0
\(447\) 21.9467 1.03804
\(448\) 0 0
\(449\) 8.08587 0.381596 0.190798 0.981629i \(-0.438892\pi\)
0.190798 + 0.981629i \(0.438892\pi\)
\(450\) 0 0
\(451\) −3.97894 −0.187361
\(452\) 0 0
\(453\) −29.2635 −1.37492
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.72831 −0.361515 −0.180758 0.983528i \(-0.557855\pi\)
−0.180758 + 0.983528i \(0.557855\pi\)
\(458\) 0 0
\(459\) 24.7754 1.15641
\(460\) 0 0
\(461\) −11.8351 −0.551214 −0.275607 0.961270i \(-0.588879\pi\)
−0.275607 + 0.961270i \(0.588879\pi\)
\(462\) 0 0
\(463\) −10.0041 −0.464931 −0.232465 0.972605i \(-0.574679\pi\)
−0.232465 + 0.972605i \(0.574679\pi\)
\(464\) 0 0
\(465\) 0.857065 0.0397455
\(466\) 0 0
\(467\) 35.8158 1.65736 0.828679 0.559724i \(-0.189093\pi\)
0.828679 + 0.559724i \(0.189093\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.4995 0.668101
\(472\) 0 0
\(473\) −9.47407 −0.435618
\(474\) 0 0
\(475\) −0.0851241 −0.00390576
\(476\) 0 0
\(477\) 1.54026 0.0705237
\(478\) 0 0
\(479\) 20.2210 0.923919 0.461960 0.886901i \(-0.347146\pi\)
0.461960 + 0.886901i \(0.347146\pi\)
\(480\) 0 0
\(481\) −4.35125 −0.198400
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.3024 0.467807
\(486\) 0 0
\(487\) −1.12743 −0.0510889 −0.0255445 0.999674i \(-0.508132\pi\)
−0.0255445 + 0.999674i \(0.508132\pi\)
\(488\) 0 0
\(489\) −3.08318 −0.139426
\(490\) 0 0
\(491\) −28.3557 −1.27968 −0.639838 0.768510i \(-0.720998\pi\)
−0.639838 + 0.768510i \(0.720998\pi\)
\(492\) 0 0
\(493\) −10.2714 −0.462600
\(494\) 0 0
\(495\) 6.12860 0.275460
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 17.4975 0.783296 0.391648 0.920115i \(-0.371905\pi\)
0.391648 + 0.920115i \(0.371905\pi\)
\(500\) 0 0
\(501\) 1.95216 0.0872159
\(502\) 0 0
\(503\) 5.12359 0.228450 0.114225 0.993455i \(-0.463562\pi\)
0.114225 + 0.993455i \(0.463562\pi\)
\(504\) 0 0
\(505\) −16.4485 −0.731947
\(506\) 0 0
\(507\) −9.88929 −0.439199
\(508\) 0 0
\(509\) 3.99019 0.176862 0.0884311 0.996082i \(-0.471815\pi\)
0.0884311 + 0.996082i \(0.471815\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −0.206726 −0.00912717
\(514\) 0 0
\(515\) 28.0653 1.23671
\(516\) 0 0
\(517\) −52.9947 −2.33070
\(518\) 0 0
\(519\) 14.5573 0.638997
\(520\) 0 0
\(521\) 19.7251 0.864174 0.432087 0.901832i \(-0.357777\pi\)
0.432087 + 0.901832i \(0.357777\pi\)
\(522\) 0 0
\(523\) −20.7789 −0.908599 −0.454299 0.890849i \(-0.650110\pi\)
−0.454299 + 0.890849i \(0.650110\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.60156 0.0697649
\(528\) 0 0
\(529\) −22.6801 −0.986093
\(530\) 0 0
\(531\) −5.37868 −0.233415
\(532\) 0 0
\(533\) −2.47100 −0.107031
\(534\) 0 0
\(535\) −8.90911 −0.385175
\(536\) 0 0
\(537\) −10.3108 −0.444946
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 21.6710 0.931708 0.465854 0.884862i \(-0.345747\pi\)
0.465854 + 0.884862i \(0.345747\pi\)
\(542\) 0 0
\(543\) −11.3984 −0.489152
\(544\) 0 0
\(545\) 12.3529 0.529139
\(546\) 0 0
\(547\) 39.5206 1.68978 0.844890 0.534940i \(-0.179666\pi\)
0.844890 + 0.534940i \(0.179666\pi\)
\(548\) 0 0
\(549\) 11.9081 0.508225
\(550\) 0 0
\(551\) 0.0857046 0.00365114
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 4.12850 0.175245
\(556\) 0 0
\(557\) 18.7066 0.792625 0.396313 0.918116i \(-0.370290\pi\)
0.396313 + 0.918116i \(0.370290\pi\)
\(558\) 0 0
\(559\) −5.88358 −0.248849
\(560\) 0 0
\(561\) −25.0051 −1.05572
\(562\) 0 0
\(563\) 9.29853 0.391886 0.195943 0.980615i \(-0.437223\pi\)
0.195943 + 0.980615i \(0.437223\pi\)
\(564\) 0 0
\(565\) 2.87077 0.120774
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −21.7314 −0.911028 −0.455514 0.890229i \(-0.650545\pi\)
−0.455514 + 0.890229i \(0.650545\pi\)
\(570\) 0 0
\(571\) 18.3218 0.766745 0.383373 0.923594i \(-0.374762\pi\)
0.383373 + 0.923594i \(0.374762\pi\)
\(572\) 0 0
\(573\) −31.8986 −1.33258
\(574\) 0 0
\(575\) −1.31697 −0.0549216
\(576\) 0 0
\(577\) −0.722827 −0.0300917 −0.0150458 0.999887i \(-0.504789\pi\)
−0.0150458 + 0.999887i \(0.504789\pi\)
\(578\) 0 0
\(579\) 12.8087 0.532311
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.50332 −0.269340
\(584\) 0 0
\(585\) 3.80598 0.157358
\(586\) 0 0
\(587\) 32.7638 1.35231 0.676153 0.736761i \(-0.263646\pi\)
0.676153 + 0.736761i \(0.263646\pi\)
\(588\) 0 0
\(589\) −0.0133634 −0.000550630 0
\(590\) 0 0
\(591\) 13.9147 0.572374
\(592\) 0 0
\(593\) 13.6267 0.559580 0.279790 0.960061i \(-0.409735\pi\)
0.279790 + 0.960061i \(0.409735\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14.0640 0.575600
\(598\) 0 0
\(599\) −5.02869 −0.205467 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(600\) 0 0
\(601\) 22.4794 0.916953 0.458477 0.888706i \(-0.348395\pi\)
0.458477 + 0.888706i \(0.348395\pi\)
\(602\) 0 0
\(603\) −13.3596 −0.544045
\(604\) 0 0
\(605\) −7.89749 −0.321079
\(606\) 0 0
\(607\) 32.9787 1.33856 0.669282 0.743009i \(-0.266602\pi\)
0.669282 + 0.743009i \(0.266602\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −32.9107 −1.33143
\(612\) 0 0
\(613\) 24.7081 0.997951 0.498976 0.866616i \(-0.333710\pi\)
0.498976 + 0.866616i \(0.333710\pi\)
\(614\) 0 0
\(615\) 2.34450 0.0945394
\(616\) 0 0
\(617\) 22.7493 0.915852 0.457926 0.888990i \(-0.348593\pi\)
0.457926 + 0.888990i \(0.348593\pi\)
\(618\) 0 0
\(619\) 7.90664 0.317795 0.158897 0.987295i \(-0.449206\pi\)
0.158897 + 0.987295i \(0.449206\pi\)
\(620\) 0 0
\(621\) −3.19830 −0.128343
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.93443 −0.317377
\(626\) 0 0
\(627\) 0.208643 0.00833239
\(628\) 0 0
\(629\) 7.71474 0.307607
\(630\) 0 0
\(631\) −33.0800 −1.31689 −0.658447 0.752627i \(-0.728786\pi\)
−0.658447 + 0.752627i \(0.728786\pi\)
\(632\) 0 0
\(633\) −23.4380 −0.931576
\(634\) 0 0
\(635\) −5.10594 −0.202623
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −6.69167 −0.264718
\(640\) 0 0
\(641\) −25.9032 −1.02311 −0.511557 0.859250i \(-0.670931\pi\)
−0.511557 + 0.859250i \(0.670931\pi\)
\(642\) 0 0
\(643\) −6.69377 −0.263977 −0.131988 0.991251i \(-0.542136\pi\)
−0.131988 + 0.991251i \(0.542136\pi\)
\(644\) 0 0
\(645\) 5.58239 0.219806
\(646\) 0 0
\(647\) −9.44119 −0.371172 −0.185586 0.982628i \(-0.559418\pi\)
−0.185586 + 0.982628i \(0.559418\pi\)
\(648\) 0 0
\(649\) 22.7100 0.891444
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.3525 1.65738 0.828691 0.559706i \(-0.189086\pi\)
0.828691 + 0.559706i \(0.189086\pi\)
\(654\) 0 0
\(655\) 14.9654 0.584746
\(656\) 0 0
\(657\) −4.94543 −0.192940
\(658\) 0 0
\(659\) 0.166508 0.00648622 0.00324311 0.999995i \(-0.498968\pi\)
0.00324311 + 0.999995i \(0.498968\pi\)
\(660\) 0 0
\(661\) −18.0731 −0.702963 −0.351481 0.936195i \(-0.614322\pi\)
−0.351481 + 0.936195i \(0.614322\pi\)
\(662\) 0 0
\(663\) −15.5286 −0.603082
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.32596 0.0513412
\(668\) 0 0
\(669\) 24.3429 0.941151
\(670\) 0 0
\(671\) −50.2785 −1.94098
\(672\) 0 0
\(673\) −27.6060 −1.06413 −0.532067 0.846702i \(-0.678584\pi\)
−0.532067 + 0.846702i \(0.678584\pi\)
\(674\) 0 0
\(675\) 13.1686 0.506860
\(676\) 0 0
\(677\) −3.03680 −0.116714 −0.0583569 0.998296i \(-0.518586\pi\)
−0.0583569 + 0.998296i \(0.518586\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 34.3952 1.31803
\(682\) 0 0
\(683\) −7.28916 −0.278912 −0.139456 0.990228i \(-0.544535\pi\)
−0.139456 + 0.990228i \(0.544535\pi\)
\(684\) 0 0
\(685\) 29.4142 1.12386
\(686\) 0 0
\(687\) 7.28315 0.277870
\(688\) 0 0
\(689\) −4.03868 −0.153862
\(690\) 0 0
\(691\) 4.41348 0.167897 0.0839483 0.996470i \(-0.473247\pi\)
0.0839483 + 0.996470i \(0.473247\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.1703 0.803037
\(696\) 0 0
\(697\) 4.38106 0.165944
\(698\) 0 0
\(699\) −17.5421 −0.663505
\(700\) 0 0
\(701\) −9.55509 −0.360891 −0.180445 0.983585i \(-0.557754\pi\)
−0.180445 + 0.983585i \(0.557754\pi\)
\(702\) 0 0
\(703\) −0.0643719 −0.00242783
\(704\) 0 0
\(705\) 31.2260 1.17604
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 24.8291 0.932477 0.466239 0.884659i \(-0.345609\pi\)
0.466239 + 0.884659i \(0.345609\pi\)
\(710\) 0 0
\(711\) −2.02329 −0.0758794
\(712\) 0 0
\(713\) −0.206748 −0.00774279
\(714\) 0 0
\(715\) −16.0697 −0.600972
\(716\) 0 0
\(717\) −24.5263 −0.915950
\(718\) 0 0
\(719\) −32.7345 −1.22079 −0.610395 0.792097i \(-0.708989\pi\)
−0.610395 + 0.792097i \(0.708989\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −9.48288 −0.352672
\(724\) 0 0
\(725\) −5.45945 −0.202759
\(726\) 0 0
\(727\) −17.7398 −0.657932 −0.328966 0.944342i \(-0.606700\pi\)
−0.328966 + 0.944342i \(0.606700\pi\)
\(728\) 0 0
\(729\) 29.3160 1.08578
\(730\) 0 0
\(731\) 10.4315 0.385825
\(732\) 0 0
\(733\) 13.5257 0.499582 0.249791 0.968300i \(-0.419638\pi\)
0.249791 + 0.968300i \(0.419638\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 56.4071 2.07778
\(738\) 0 0
\(739\) 17.3968 0.639952 0.319976 0.947426i \(-0.396325\pi\)
0.319976 + 0.947426i \(0.396325\pi\)
\(740\) 0 0
\(741\) 0.129571 0.00475992
\(742\) 0 0
\(743\) −7.79349 −0.285915 −0.142958 0.989729i \(-0.545661\pi\)
−0.142958 + 0.989729i \(0.545661\pi\)
\(744\) 0 0
\(745\) −25.0066 −0.916172
\(746\) 0 0
\(747\) 12.3460 0.451715
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 24.0634 0.878087 0.439043 0.898466i \(-0.355317\pi\)
0.439043 + 0.898466i \(0.355317\pi\)
\(752\) 0 0
\(753\) −20.8294 −0.759066
\(754\) 0 0
\(755\) 33.3436 1.21350
\(756\) 0 0
\(757\) 48.9399 1.77875 0.889376 0.457177i \(-0.151140\pi\)
0.889376 + 0.457177i \(0.151140\pi\)
\(758\) 0 0
\(759\) 3.22796 0.117167
\(760\) 0 0
\(761\) 27.3401 0.991079 0.495540 0.868585i \(-0.334970\pi\)
0.495540 + 0.868585i \(0.334970\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6.74797 −0.243973
\(766\) 0 0
\(767\) 14.1033 0.509242
\(768\) 0 0
\(769\) −20.9324 −0.754842 −0.377421 0.926042i \(-0.623189\pi\)
−0.377421 + 0.926042i \(0.623189\pi\)
\(770\) 0 0
\(771\) 5.42671 0.195438
\(772\) 0 0
\(773\) 46.7545 1.68164 0.840822 0.541312i \(-0.182072\pi\)
0.840822 + 0.541312i \(0.182072\pi\)
\(774\) 0 0
\(775\) 0.851260 0.0305782
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.0365556 −0.00130974
\(780\) 0 0
\(781\) 28.2537 1.01100
\(782\) 0 0
\(783\) −13.2584 −0.473817
\(784\) 0 0
\(785\) −16.5211 −0.589662
\(786\) 0 0
\(787\) 5.76023 0.205330 0.102665 0.994716i \(-0.467263\pi\)
0.102665 + 0.994716i \(0.467263\pi\)
\(788\) 0 0
\(789\) −43.7732 −1.55837
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −31.2239 −1.10879
\(794\) 0 0
\(795\) 3.83194 0.135905
\(796\) 0 0
\(797\) 24.6110 0.871766 0.435883 0.900003i \(-0.356436\pi\)
0.435883 + 0.900003i \(0.356436\pi\)
\(798\) 0 0
\(799\) 58.3505 2.06429
\(800\) 0 0
\(801\) −1.25405 −0.0443097
\(802\) 0 0
\(803\) 20.8807 0.736864
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.80409 0.204314
\(808\) 0 0
\(809\) −30.9653 −1.08868 −0.544341 0.838864i \(-0.683220\pi\)
−0.544341 + 0.838864i \(0.683220\pi\)
\(810\) 0 0
\(811\) −47.3652 −1.66322 −0.831609 0.555362i \(-0.812580\pi\)
−0.831609 + 0.555362i \(0.812580\pi\)
\(812\) 0 0
\(813\) 0.186756 0.00654983
\(814\) 0 0
\(815\) 3.51305 0.123057
\(816\) 0 0
\(817\) −0.0870409 −0.00304518
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.09335 −0.0381582 −0.0190791 0.999818i \(-0.506073\pi\)
−0.0190791 + 0.999818i \(0.506073\pi\)
\(822\) 0 0
\(823\) 33.9949 1.18499 0.592493 0.805575i \(-0.298144\pi\)
0.592493 + 0.805575i \(0.298144\pi\)
\(824\) 0 0
\(825\) −13.2907 −0.462723
\(826\) 0 0
\(827\) 16.5948 0.577059 0.288529 0.957471i \(-0.406834\pi\)
0.288529 + 0.957471i \(0.406834\pi\)
\(828\) 0 0
\(829\) 1.51910 0.0527605 0.0263803 0.999652i \(-0.491602\pi\)
0.0263803 + 0.999652i \(0.491602\pi\)
\(830\) 0 0
\(831\) 39.2985 1.36325
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −2.22433 −0.0769762
\(836\) 0 0
\(837\) 2.06731 0.0714565
\(838\) 0 0
\(839\) 41.9564 1.44849 0.724247 0.689541i \(-0.242188\pi\)
0.724247 + 0.689541i \(0.242188\pi\)
\(840\) 0 0
\(841\) −23.5033 −0.810459
\(842\) 0 0
\(843\) −25.7442 −0.886676
\(844\) 0 0
\(845\) 11.2681 0.387634
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −28.5345 −0.979303
\(850\) 0 0
\(851\) −0.995912 −0.0341394
\(852\) 0 0
\(853\) −31.6644 −1.08417 −0.542084 0.840324i \(-0.682365\pi\)
−0.542084 + 0.840324i \(0.682365\pi\)
\(854\) 0 0
\(855\) 0.0563052 0.00192560
\(856\) 0 0
\(857\) 45.4789 1.55353 0.776764 0.629792i \(-0.216860\pi\)
0.776764 + 0.629792i \(0.216860\pi\)
\(858\) 0 0
\(859\) 21.2872 0.726309 0.363154 0.931729i \(-0.381700\pi\)
0.363154 + 0.931729i \(0.381700\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.77970 0.162703 0.0813514 0.996685i \(-0.474076\pi\)
0.0813514 + 0.996685i \(0.474076\pi\)
\(864\) 0 0
\(865\) −16.5870 −0.563975
\(866\) 0 0
\(867\) 3.14667 0.106866
\(868\) 0 0
\(869\) 8.54278 0.289794
\(870\) 0 0
\(871\) 35.0299 1.18694
\(872\) 0 0
\(873\) 5.94013 0.201043
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −49.9215 −1.68573 −0.842864 0.538126i \(-0.819132\pi\)
−0.842864 + 0.538126i \(0.819132\pi\)
\(878\) 0 0
\(879\) −14.0046 −0.472362
\(880\) 0 0
\(881\) −6.69012 −0.225396 −0.112698 0.993629i \(-0.535949\pi\)
−0.112698 + 0.993629i \(0.535949\pi\)
\(882\) 0 0
\(883\) −39.3012 −1.32259 −0.661295 0.750125i \(-0.729993\pi\)
−0.661295 + 0.750125i \(0.729993\pi\)
\(884\) 0 0
\(885\) −13.3813 −0.449809
\(886\) 0 0
\(887\) −52.5777 −1.76539 −0.882693 0.469950i \(-0.844272\pi\)
−0.882693 + 0.469950i \(0.844272\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −21.0278 −0.704457
\(892\) 0 0
\(893\) −0.486877 −0.0162927
\(894\) 0 0
\(895\) 11.7484 0.392706
\(896\) 0 0
\(897\) 2.00462 0.0669325
\(898\) 0 0
\(899\) −0.857065 −0.0285847
\(900\) 0 0
\(901\) 7.16056 0.238553
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.9876 0.431722
\(906\) 0 0
\(907\) 31.3793 1.04193 0.520967 0.853577i \(-0.325571\pi\)
0.520967 + 0.853577i \(0.325571\pi\)
\(908\) 0 0
\(909\) −9.48383 −0.314559
\(910\) 0 0
\(911\) 5.20500 0.172449 0.0862247 0.996276i \(-0.472520\pi\)
0.0862247 + 0.996276i \(0.472520\pi\)
\(912\) 0 0
\(913\) −52.1273 −1.72516
\(914\) 0 0
\(915\) 29.6255 0.979390
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −24.7249 −0.815599 −0.407799 0.913072i \(-0.633704\pi\)
−0.407799 + 0.913072i \(0.633704\pi\)
\(920\) 0 0
\(921\) −0.819640 −0.0270080
\(922\) 0 0
\(923\) 17.5461 0.577536
\(924\) 0 0
\(925\) 4.10054 0.134825
\(926\) 0 0
\(927\) 16.1819 0.531482
\(928\) 0 0
\(929\) −54.1957 −1.77810 −0.889051 0.457808i \(-0.848635\pi\)
−0.889051 + 0.457808i \(0.848635\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −26.7173 −0.874684
\(934\) 0 0
\(935\) 28.4914 0.931768
\(936\) 0 0
\(937\) 24.7937 0.809974 0.404987 0.914323i \(-0.367276\pi\)
0.404987 + 0.914323i \(0.367276\pi\)
\(938\) 0 0
\(939\) −41.1506 −1.34290
\(940\) 0 0
\(941\) 15.9574 0.520198 0.260099 0.965582i \(-0.416245\pi\)
0.260099 + 0.965582i \(0.416245\pi\)
\(942\) 0 0
\(943\) −0.565560 −0.0184172
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.9098 −0.776965 −0.388482 0.921456i \(-0.627001\pi\)
−0.388482 + 0.921456i \(0.627001\pi\)
\(948\) 0 0
\(949\) 12.9673 0.420937
\(950\) 0 0
\(951\) 20.8173 0.675046
\(952\) 0 0
\(953\) 55.6927 1.80406 0.902032 0.431668i \(-0.142075\pi\)
0.902032 + 0.431668i \(0.142075\pi\)
\(954\) 0 0
\(955\) 36.3460 1.17613
\(956\) 0 0
\(957\) 13.3813 0.432558
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.8664 −0.995689
\(962\) 0 0
\(963\) −5.13681 −0.165531
\(964\) 0 0
\(965\) −14.5945 −0.469815
\(966\) 0 0
\(967\) 17.1200 0.550541 0.275271 0.961367i \(-0.411233\pi\)
0.275271 + 0.961367i \(0.411233\pi\)
\(968\) 0 0
\(969\) −0.229729 −0.00737995
\(970\) 0 0
\(971\) 12.2317 0.392534 0.196267 0.980551i \(-0.437118\pi\)
0.196267 + 0.980551i \(0.437118\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −8.25378 −0.264333
\(976\) 0 0
\(977\) −48.6047 −1.55500 −0.777500 0.628883i \(-0.783513\pi\)
−0.777500 + 0.628883i \(0.783513\pi\)
\(978\) 0 0
\(979\) 5.29487 0.169225
\(980\) 0 0
\(981\) 7.12240 0.227401
\(982\) 0 0
\(983\) −1.39481 −0.0444875 −0.0222437 0.999753i \(-0.507081\pi\)
−0.0222437 + 0.999753i \(0.507081\pi\)
\(984\) 0 0
\(985\) −15.8547 −0.505174
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.34663 −0.0428203
\(990\) 0 0
\(991\) 5.13504 0.163120 0.0815600 0.996668i \(-0.474010\pi\)
0.0815600 + 0.996668i \(0.474010\pi\)
\(992\) 0 0
\(993\) −27.9031 −0.885477
\(994\) 0 0
\(995\) −16.0248 −0.508022
\(996\) 0 0
\(997\) 24.6680 0.781244 0.390622 0.920551i \(-0.372260\pi\)
0.390622 + 0.920551i \(0.372260\pi\)
\(998\) 0 0
\(999\) 9.95826 0.315065
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8036.2.a.j.1.4 5
7.6 odd 2 1148.2.a.e.1.2 5
28.27 even 2 4592.2.a.bd.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.a.e.1.2 5 7.6 odd 2
4592.2.a.bd.1.4 5 28.27 even 2
8036.2.a.j.1.4 5 1.1 even 1 trivial